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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 51, Iss. 18 — Jun. 20, 2012
  • pp: 4087–4091

Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils

Virendra N. Mahajan  »View Author Affiliations


Applied Optics, Vol. 51, Issue 18, pp. 4087-4091 (2012)
http://dx.doi.org/10.1364/AO.51.004087


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Abstract

In a recent paper, we considered the classical aberrations of an anamorphic optical imaging system with a rectangular pupil, representing the terms of a power series expansion of its aberration function. These aberrations are inherently separable in the Cartesian coordinates (x,y) of a point on the pupil. Accordingly, there is x-defocus and x-coma, y-defocus and y-coma, and so on. We showed that the aberration polynomials orthonormal over the pupil and representing balanced aberrations for such a system are represented by the products of two Legendre polynomials, one for each of the two Cartesian coordinates of the pupil point; for example, Ll(x)Lm(y), where l and m are positive integers (including zero) and Ll(x), for example, represents an orthonormal Legendre polynomial of degree l in x. The compound two-dimensional (2D) Legendre polynomials, like the classical aberrations, are thus also inherently separable in the Cartesian coordinates of the pupil point. Moreover, for every orthonormal polynomial Ll(x)Lm(y), there is a corresponding orthonormal polynomial Ll(y)Lm(x) obtained by interchanging x and y. These polynomials are different from the corresponding orthogonal polynomials for a system with rotational symmetry but a rectangular pupil. In this paper, we show that the orthonormal aberration polynomials for an anamorphic system with a circular pupil, obtained by the Gram–Schmidt orthogonalization of the 2D Legendre polynomials, are not separable in the two coordinates. Moreover, for a given polynomial in x and y, there is no corresponding polynomial obtained by interchanging x and y. For example, there are polynomials representing x-defocus, balanced x-coma, and balanced x-spherical aberration, but no corresponding y-aberration polynomials. The missing y-aberration terms are contained in other polynomials. We emphasize that the Zernike circle polynomials, although orthogonal over a circular pupil, are not suitable for an anamorphic system as they do not represent balanced aberrations for such a system.

© 2012 Optical Society of America

OCIS Codes
(010.7350) Atmospheric and oceanic optics : Wave-front sensing
(110.0110) Imaging systems : Imaging systems
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(220.0220) Optical design and fabrication : Optical design and fabrication
(220.1010) Optical design and fabrication : Aberrations (global)

ToC Category:
Imaging Systems

History
Original Manuscript: April 2, 2012
Manuscript Accepted: April 16, 2012
Published: June 14, 2012

Citation
Virendra N. Mahajan, "Orthonormal aberration polynomials for anamorphic optical imaging systems with circular pupils," Appl. Opt. 51, 4087-4091 (2012)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-51-18-4087

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